Abstract
We study the Dirichlet problem for the complex Monge–Amp`ere
operator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some regularity estimates for a large class of complex
Monge–Amp`ere measures. This result is optimal in the unit disk, as boundary
functions with b-pluripolar discontinuity then coincide with functions that are
continuous almost everywhere. We also show that neither of these properties
of the boundary function – being continuous almost everywhere or having discontinuities forming a b-pluripolar set – are necessary conditions in order to
establish uniqueness and continuity of the solution in higher dimensions. In
particular, there are situations where it is enough to prescribe the boundary
behavior at a set of arbitrarily small Lebesgue measure.
operator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some regularity estimates for a large class of complex
Monge–Amp`ere measures. This result is optimal in the unit disk, as boundary
functions with b-pluripolar discontinuity then coincide with functions that are
continuous almost everywhere. We also show that neither of these properties
of the boundary function – being continuous almost everywhere or having discontinuities forming a b-pluripolar set – are necessary conditions in order to
establish uniqueness and continuity of the solution in higher dimensions. In
particular, there are situations where it is enough to prescribe the boundary
behavior at a set of arbitrarily small Lebesgue measure.
Original language | English |
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Journal | Indiana University Mathematics Journal |
Publication status | Accepted/In press - 2023 |
Subject classification (UKÄ)
- Mathematical Analysis